Abstract This project deals with the study of multi-object auctions with multi-dimensional signals. The recent spurt in interest in the study of multiple-object auctions is largely the result of large-scale privatization of socially held assets by governments. In light of the emerging view that many results from the mechanism design and information economics literature fail to extend to multiple-object multi-dimensional settings, and in consideration of the influence economic theory has had on the recent design of a number of these privatization auctions, it is of some importance to understand which of the results from the one-good one-dimensional signal auction literature, on which economists base most of their intuition, are sensitive to dimensionality. This is especially germane since the multi-object, multi-dimensional case is by far the more common. Milgrom and Weber (1982) provide a number of important results on the revenue-ranking of various auction forms under incomplete information. However, their analysis is limited to the case in which the seller has but a single unit of an indivisible good, and the signals received by the bidders are one-dimensional. As a first step beyond the Milgrom-Weber framework, and in order to get a feel for the underlying economics, we derive the equilibrium of a Vickrey auction in the simplest possible extension of the single-unit environment. That is, we consider a homogeneous-good, two-unit setup wherein there are but two agents. The agents receive affiliated signals regarding the value of the good and there is a common-value component to the agents' valuations. Remarkably, in this very simple 2x2 environment, we are able to demonstrate the failure of the linkage principle. The significance of this finding is highlighted by the recent design of the spectral auction. It is widely agreed by the experts involved in the process that the choice to include an open-auction format in the FCC auction was largely influenced by the faith placed (now perhaps misplaced) in the linkage principle. We continue this line of research by conducting a careful study of the equilibria and their properties in multi-object/multi-dimensional auction environments. There are numerous questions we focus on. Does the linkage principle hold for multi-unit first-price auctions? Is the multi-unit ascending auction efficient? What are its equilibrium strategies? How do its revenues compare to those in the multi-unit Vickrey auction? What are the equilibrium strategies there?